Maths WorkshopMaths problems with full solutions

Mean Value Theorem

27 problems · Rolle's theorem, the mean value theorem and its consequences.

Ex.17The mean value theorem
#mean value theorem #Lagrange #existence
Mean Value Theorem
High school
High school
Ex.18Rolle's theorem on a polynomial
#Rolle #polynomial #existence
Mean Value Theorem
High school
High school
Ex.43Mean value theorem for $f(x)=x^3$
#mean value theorem #cubic #existence
Mean Value Theorem
High school
High school
Ex.44Summary of the mean value theorem
#summary #MVT #Rolle
Mean Value Theorem
High school
High school
Ex.211Rolle's theorem on a quartic
#Rolle's theorem #polynomials
Mean Value Theorem
L1
L1
Ex.349Sine is 1-Lipschitz
#mean value theorem #inequalities
Mean Value Theorem
L1
L1
Ex.377The mean value theorem and an arctangent inequality
#mean value theorem #inequalities #arctangent
Mean Value Theorem
L2
L2
Ex.378Two-sided bounds for the logarithm
#mean value theorem #inequalities #logarithm
Mean Value Theorem
L1
L1
Ex.616A function with zero derivative is constant
#consequence #constant function #zero derivative
Mean Value Theorem
L1
L1
Ex.617Sign of the derivative and monotonicity
#monotonicity #derivative sign #consequence
Mean Value Theorem
L1
L1
Ex.721Fixed point and a derivative bounded by one
#fixed points #mean value theorem #intermediate value theorem
Mean Value Theorem
L2
L2
Ex.762If f′ tends to ℓ, then f(x+1) − f(x) tends to ℓ
#mean value theorem #limits at infinity
Mean Value Theorem
L1
L1
Ex.788The limit of the derivative theorem
#mean value theorem #derivative at a point #extension
Mean Value Theorem
L2
L2
Ex.1268An arctangent inequality
#mean value theorem #inequality #arctan
Mean Value Theorem
L1
L1
Ex.1269An exponential inequality
#mean value theorem #inequality #exponential
Mean Value Theorem
L1
L1
Ex.1270A logarithm inequality via the MVT
#mean value theorem #inequality #logarithm
Mean Value Theorem
L1
L1
Ex.1271Bounding $f(b)-f(a)$
#MVT #bound #interval
Mean Value Theorem
L1
L1
Ex.1272Small-increment estimate
#MVT #small increments #estimate
Mean Value Theorem
L1
L1
Ex.1273Bounded derivative implies Lipschitz
#Lipschitz #bounded derivative #MVT
Mean Value Theorem
L1
L1
Ex.1274Rolle and the sign of $f'$
#Rolle #sign change #derivative
Mean Value Theorem
L1
L1
Ex.1275Rolle on a polynomial with three roots
#Rolle #multiple roots #derivative
Mean Value Theorem
L1
L1
Ex.1276Uniqueness of a root
#Rolle #uniqueness #contrapositive
Mean Value Theorem
L1
L1
Ex.1953Cauchy's extended mean value theorem
#Cauchy #extended MVT #ratio
Mean Value Theorem
L1
L1
Ex.1954Darboux's theorem on derivatives
#Darboux #intermediate value #derivatives
Mean Value Theorem
M1
M1
Ex.1955Second-order Taylor formula
#Taylor #MVT #second order
Mean Value Theorem
L1
L1
Ex.1956Iterated Rolle
#Rolle #higher derivatives #iterated
Mean Value Theorem
L1
L1
Ex.2075Legendre polynomials have all their roots in (-1, 1)
#Rolle's theorem #polynomials #induction #Legendre
Mean Value Theorem
L2
L2